# Full text of "MAGIC IN THE FOURTH DIMENSION"

## See other formats

```STOP

Early Journal Content on JSTOR, Free to Anyone in the World

the world by JSTOR.

Known as the Early Journal Content, this set of works include research articles, news, letters, and other
writings published in more than 200 of the oldest leading academic journals. The works date from the
mid-seventeenth to the early twentieth centuries.

We encourage people to read and share the Early Journal Content openly and to tell others that this
resource exists. People may post this content online or redistribute in any way for non-commercial
purposes.

journal-content .

JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people
discover, use, and build upon a wide range of content through a powerful research and teaching
platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit
contact support@jstor.org.

CRITICISMS AND DISCUSSIONS. 309

To bring these magical squares to the surface the squares of
each set of parallel squares may be permuted as follows:

Original order 1, 2, 3, 4, 5, 6,

Permuted order 3, 2, 1, 6, 5, 4.

The result is the final cube shown in the beginning of this article.

The above permutation is subject to two conditions. The sev-
eral sets of parallel squares must all be permuted in the same man-
ner. Any two parallel squares which in the original cube are located
on opposite sides of the middle plane of the cube and at an equal
distance from it, in the permuted cube must be located on opposite
sides of the middle plane of the cube and at an equal distance from it.
These conditions are for the protection of the diagonals.

John Worthington.

MAGIC IN THE FOURTH DIMENSION.

Definition of terms : Row is a general term ; rank denotes a hori-
zontal right-to-left row; file a row from front to back; and column
a vertical row in a cube — not used of any horizontal dimension.

If n 2 numbers of a given series can be grouped so as to form a

magic square and n such squares be so placed as to constitute a

magic cube, why may we not go a step further and group n cubes

in relations of the fourth dimension ? In a magic square containing

the natural series 1 . . . w 2 the summation is— — ' ; in a magic cube

fl( fl^-i— T ^

with the series i...» 3 it is — - — ^— -; and in an analogous fourth-

dimension construction it naturally will be — - — — — .

With this idea in mind I have made some experiments, and the
results are interesting. The analogy with squares and cubes is not
perfect, for rows of numbers can be arranged side by side to repre-
sent a visible square, squares can be piled one upon another to make
a visible cube, but cubes cannot be so combined in drawing as to
picture to the eye their higher relations. My expectation a priori
was that some connection or relation, probably through some form
of diagonal-of-diagonal, would be found to exist between the cubes
containing the n* terms of a series. This particular feature did ap-
pear in the cases where n was odd. Here is how it worked out:

I. When n is odd.
1. Let w=3, then 8=123. — The natural series 1. . .81 was di-
vided into three sub-series such that the sum of each would be

3io

THE MONIST.

one-third the sum of the whole. In dealing with any such series
when n is odd there will be n sub-series, each starting with one of
the first w numbers, and the difference between successive terms will
be w+i, except after a multiple of n, when the difference is i. In
the present case the three sub-series begin respectively with i, 2, 3,
and the first is 1 5 9 10 14 18 19 23 27 28 32 36 37 41 45 46 50 54
55 59 63 64 68 72 73 yy 81. These numbers were arranged in
three squares constituting a magic cube, and the row of squares
so formed was flanked on right and left by similar rows formed from
the other two sub-series (see Fig. 1).

It is not easy — perhaps it is not possible — to make an abso-
lutely perfect cube of 3. These are not perfect, yet they have many

25

38

60

28

77

18

67

8

48

33

79

11

72

1

SO

21

40

62

65

6

52

23

45

55

35

75

13

29

78

16

68

9

46

26

39

58

70

2

5i

19

41

63

31

80

12

24

43

56

36

73

14

66

4

53

69

7

47

27

37

59

30

76

17

20

42

61

32

81

10

71

3

49

34

74

15

64

5

54

22

44

57

Fig. 1. (3')

striking features. Taking the three cubes separately we find that
in each all the "straight" dimensions — rank, file and column — have
the proper footing, 123. In the middle cube there are two plane
diagonals having the same summation, and in cubes I and III one
each. In cube II four cubic diagonals and four diagonals of vertical
squares are correct ; I and III each have one cubic diagonal and one
vertical-square diagonal.

So much for the original cubes; now for some combinations.
The three squares on the diagonal running down from left to right
will make a magic cube with rank, file, column, cubic diagonals,
two plane diagonals and four vertical-square diagonals (37 in all)
correct. Two other cubes can be formed by starting with the top
squares of II and III respectively and following the "broken diag-

CRITICISMS AND DISCUSSIONS.

3"

onals" running downward to the right. In each of these S occurs
at least 28 times (in 9 ranks, 9 files, 9 columns and one cubic diag-
onal). Various other combinations may be found by taking the
squares together in horizontal rows and noting how some columns
and assorted diagonals have the proper summation, but the most
important and significant are those already pointed out. In all the
sum 123 occurs over 200 times in this small figure.

317

473

604

10

161

T92

348

479

510

36

67

223

354

385

536

567

98

229

260

4"

442

598

104

135

286

110

■36

292

448

579

6to

n

167

323

454

485

5"

42

198

329

360

386

542

73

204

235

261

4'7

573

79

423

554

85

236

267

298

429

585

ITI

1+2

'73

304

460

61 1

■ 7

48

179

335

486

517

548

54

210

361

392

21 r

367

398

529

60

86

242

273

404

560

586

"7

148

279

435

461

6'7

23

154

3'0

336

492

523

29

185

504

35

186

342

498

379

535

6i

217

373

254

410

561

92

248

129

285

436

592

'23

4

160

3"

467

623

606

12

168

324

455

481

512

43

'99

330

356

387

543

74

205

231

262

418

574

80

106

137

293

449

580

299

430

58l

112

'43

(74

305

456

612

18

49

180

33'

487

5'8

549

55

206

362

393

424

555

81

237

268

87

243

274

405

556

587

118

149

280

43'

462

618

24

'55

306

337

493

524

30

181

212

368

399

530

56

380

531

62

218

374

255

406

562

93

249

130

281

437

593

124

5

'56

3'2

468

624

505

3'

'87

343

499

■93

349

480

506

37

68

224

355

38>

537

568.

99

230

256

412

443

599

105

131

287

3'8

474

605

6

162

'75

3t"

457

613

'9

50

176

332

488

5'9

550

5'

207

363

394

425

55'

82

238

269

300

426

582

"3

'44

588

l'9

150

276

432

463

619

25

151

307

338

494

525

26

182

213

309

400

526

57

88

244

275

40J

557

25 r

407

563

94

250

126

282

438

594

125

1 ,

'57

3'3

469

625

501

32

188

344

500

376

532

63

219

375
38

€9

225

351

382

538

569

100

226

257

4U

444

600

101

132

288

319

475

601

7

■63

'94

350

476

507

482

5J3

44

200

326

357

388

544

75

201

232

263

419

575

76

107

138

294

450

576

607

13

169

325

45'

464

620

21

rS2

308

339

495

521

27

183

214

370

396

527

58

89

245

271

402

558

589

120

146

277

433

127

283

439

595

121

2

158

J'4

470

621

502

33

189

345

496

377

533

64

220

37'

252

408

564

95

246

570

96

227

258

414

445

596

102

'33

289

320

47'

602

8

164

'95

346

477

508

39

7P

221

352

383

539

358

389

545

71

202

233

264

420

57'

77

108

'39

295

446

577

608

'4

'70

321

452

483

5'4

45

196

327

46

177

333

489

520

546

52

208

364

395

421

552

83

239

270

296

427

583

"4

'45

171

302

458

614

20

3

159

3J5

466

622

503

34

190

34'

497

378

534

65

216

372

253

409

565

9'

247

128

284

440

59'

122

441

597

'03

134

290

3.6

472

603

9

'65

191

347

478

509

40

66

222

353

384

540

566

97

288

259

4'5

234

265

416

572

78

109

140

291

447

578

609

'5

166

322

453

484

5'5

4'

197

328

359

390

54'

72

203

547

53

209

365

391

422

553

84

240

266

297

428

584

"5

14'

172

303

459

615

16

47

■ 78

334

490

516

340

491

5«

28

■84

215

366

397

528

59

90

241

272

403

559

590

116

'47

278

434

465

616

22

'53

30S

Fig. 2. (s 4 )

One most interesting fact remains to be noticed. While the
three cubes were constructed separately and independently the figure
formed by combining them is an absolutely perfect square of 9, with
a summation of 369 in rank, file and corner diagonal (besides all
"broken" diagonals running downward to the right), and a perfect

312 THE MONIST.

balancing of complementary numbers about the center. Any such
pair, taken with the central number 41, gives us the familiar sum 123,
and this serves to bind the whole together in a remarkable manner.

2. Let w=5, then 8=1565. — In Fig. 2 is represented a group
of 5-cubes each made up of the numbers in a sub-series of the nat-
ural series 1 . . .625. In accordance with the principle stated in a pre-
vious paragraph the central sub-series is 1 7 13 19 25 26 32 ... 625,
and the other four can easily be discovered by inspection. Each of
the twenty-five small squares has the summation 1565 in rank, file,
corner diagonal and broken diagonals, twenty times altogether in
each square, or 500 times for all.

Combining the five squares in col. I we have a cube in which
all the 75 "straight" rows (rank, file and vertical column), all the
horizontal diagonals and three of the four cubic diagonals foot up
1565. In cube III all the cubic diagonals are correct. Each cube
also has seven vertical-square diagonals with the same summation.
Taking together the squares in horizontal rows we find certain
diagonals having the same sum, but the columns do not. The five
squares in either diagonal of the large square, however, combine to
produce almost perfect cubes, with rank, file, column and cubic
diagonals all correct, and many diagonals of vertical squares.

A still more remarkable fact is that the squares in the broken
diagonals running in either direction also combine to produce cubes
as nearly perfect as those first considered. Indeed, the great square
seems to be an enlarged copy of the small squares, and where the
cells in the small ones unite to produce S the corresponding squares
in the large figure unite to produce cubes more or less perfect.
Many other combinations are discoverable, but these are sufficient
to illustrate the principle, and show the interrelations of the cubes
and their constituent squares. The summation 1565 occurs in this
figure not less than 1400 times.

The plane figure containing the five cubes (or twenty-five
squares) is itself a perfect square with a summation of 7825
for every rank, file, corner or broken diagonal. Furthermore all
complementary pairs are balanced about the center, as in Fig. 1.
Any square group of four, nine or sixteen of the small squares is
magic, and if the group of nine is taken at the center it is "perfect."
It is worthy of notice that all the powers of « above the first lie in
the middle rank of squares, and that all other multiples of « are
grouped in regular relations in the other ranks and have the same

CRITICISMS AND DISCUSSIONS.

313

grouping in all the squares of any given rank. The same is true
of the figure illustrating 7*, which is to be considered next.

3. Let n=7, then 8=8407. — This is so similar in all its prop-
erties to the 5-construction just discussed that it hardly needs sep-
arate description. It is more nearly perfect in all its parts than the
5*, having a larger proportion of its vertical-square diagonals cor-
rect. Any square group of four, nine, sixteen, twenty-five or thirty-
six small squares is magic, and if the group of nine or twenty-five

I

255

254

4

248

10

11

245

240

18

19

237

25

231

230

28

252

6

7

249

13

243

242

16

21

235

234

24

228

30

31

225

8

250

251

5

241

15

14

244

233

23

22

236

32

226

227

29

253

3

2

256

12

246

247

9

20

238

239

17

229

27

26

232

224

34

35

221

4i

215

214

44

49

207

206

52

200

58

59

197

37

219

218

40

212

46

47

209

204

54

55

201

61

195

194

64

217

39

38

220

48

210

211

45

56

202

203

53

193

63

62

196

36

222

223

33

213

43

42

216

205

51

50

208

60

198

199

57

192

66

67

189

73

183

182

76

81

175

174

84

168

90

91

165

69

187

186

72

180

78

79

177

172

86

87

169

93

163

162

96

185

71

70

188

80

178

179

77

88

170

171

85

161

95

94

164

68

190

191

65

181

75

74

184

173

83

82

176

92

166

167

89

97

159

158

100

152

106

107

149

144

114

115

141

121

135

134

124

156

102

103

153

109

147

146

112

«7

139

138

120

132

126

127

129

104

154

155

101

145

iti

1 10

148

137

119

118

140

128

130

131

125

157

99

98

160

108

150

151

105

116

142

143]

113

133

123

122

136

Fig. 3- (4 4 )

be taken at the center of the figure it is "perfect." The grouping
of multiples and powers of n is very similar to that already described
for 5*.

II. When n is even.

I. Let «=4, then 8=514. — The numbers may be arranged in
either of two ways. If we take the diagram for the 4-cube as

II

III

1

1295

1294

3

1292

6

1278

20

21

1276

23

"73

37

1259

1258

39

1256

42

1290

8

■288

1287

11

7

25

1271

27

28

1268

1272

'254

44

.1252

1251

47

43

1284

1283

'5

16

>4

1279

31

32

1264

1263

1265

36

1248

1247

5'

52

50

'243

'3

'7

1281

1282

1280

18

1266

1262

34

33

35

1261

49

53

1245

1246

1244

54

12

1286

9

10

1289

1285

1267

29

1270

1269

26

30

48

1250

45

46

1253

1249

1291

3

4

'293

5

1296

24

1277

■275

22

•274

'9

1255

38

40

'257

4'

1260

1188

no

in

1 186

"3

1 183

127

1 169

1168

129

1166

'32

"52

146

'47

"50

'49

"47

"5

1 181

"7

118

1 178

1 182

1 164

134

1 162

1 161

'37

'33

'5'

"45

'53

154

1142

"46

121

122

1 174

"73

"75

126

1158

"67

141

142

140

"53

'57

■58

1 138

"37

"39

162

1176

1 172

124

123

125

1171

■39

'43

"55

"56

"54

144

1 140

"36

160

'59

161

"35

1177

119

1180

1 179

116

120

'38

u6o

'35

•36

"63

"59

1 141

'55

"44

"43

152

•56

114

1187

1185

112

1184

I09

1165

128

130

1167

'3'

1 170

'5°

"5'

"49

148

1 148

'45

217

1079

1078

219

1076

222

1062

236

237

1060

239

1057

253

1043

1042

255

1040

258

1074

224

1072

1071

227

223

241

i°55

243

244

1052

1056

1038

260

1036

1035

263

259

1068

1067

231

232

230

1063

247

248

1048

1047

1049

252

1032

1031

267

268

266

1027

229

233

1065

1066

1064

234

1050

1046

250

249

251

'045

265

269

1029

1030

1028

270

228

1070

225

226

1073

I069

105 1

245

1054

">53

242

246

264

1034

261

262

1037

>°33

1075

218

220

1077

221

1080

240

106 1

1059

238

1058

235

1039

254

256

1041

257

1044

865

43t

43°

867

428

870

4H

884

885

4U

887

409

901

395

394

903

392

906

426

872

424

423

875

871

889

407

891

892

404

408

390

908

388

387

9"

907

420

419

879

880

878

415

895

896

400

399

401

900

384

&S

915

916

914

379

877

881

417

418

416

882

402

398

898

897

899

397

9'3

9'7

381

382

380

918

876

422

873

874

425

421

403

'893

406

405

890

894

912

386

909

910

389

385

427

866

868

429

869

432

888

4'3

411

886

410

883

39'-

902

904

393

9°5

396

864

434

435

862

437

859

451

845

844

453

842

456

828

470

47'

826

473

823

439

857

441

442

854

858

840

458

838

837

461

457

475

821

477

478

818

822

445

446

850

849

8S«

450

834

833

465

466

464

829

481

482

814

8>3

815

486

852

848

448

447

449

847

463

467

83'

832

830

468

816

812

484

483

485

811

853

443

856

855

440

444

462

836

459

460

839

835

8i7

479

820

819

476

480

438

863

861

436

860

433

841

452

454

843

455

846

474

827

825

472

824

469

756

542

543

754

545

751

559

737

736

56i

734

564

720

578

579

7'8

58i

7'5

547

749

549

55°

746

750

732

566

73°

729

569

565

583

7'3

585

586

710

7'4

553

554

742

741

743

558

726

725

573

574

572

721

589

59°

706

7°5

707

594

744

740

556

555

557

739

57"

575

723

724

722

576

708

704

592

59'

593

703

745

55"

748

747

548

552

57°

728

567

568

73"

727

709

587

712

7"

584

588

546

755-

753

544

752

541

733

560

562

735

563

738

582

7W

7'7

580

7J6

577

Fig. 4, First Part. (6*: 8=3891)

IV

VI

122\$

71

70

1227

68

1230

1224

74

75

1222

77

1219

120^

92

93

1204

95

1201

r>r>

1232

64

f>3

'235

■23'

79

1217

81

82

1214

1218

97

"99

99

100

119C

1200

fio

59

'239

1240

1238

55

85

86

1210

1209

[211

90

>°3

104

1192

1191

"93

108

'237

124!

57

58

56

1242

1212

1208

88

87

89

1207

"94

1190

106

■°5

107

1189

1236

62

<m

1234

65

61

1213

83

1216

1215

80

84

"95

101

1 198

"97

98

102

(>7

1226

1228

69

1229

72

78

1223

1221

7*

1220

73

96

1205

1203

94

1202

91

180

1 1 18

1 1 19

178

1121

'75

181

'"5

1 114

'83

1 1 12

186

'99

1097

1096

20 r

IO94

204

1123

'73

1125

I 126

170

'74

11 10

188

1108

1107

KJI

187

1092

206

1090

1089

209

205

1129

1130

166

"•'5

167

"34

1 104

1103

'95

196

194

1099

1086

1085

213

214

212

1081

168

164

1132

"3'

"33

""'3

'93

'97

MOI

1 102

MOO

198

211

2'5

1083

1084

I082

216

169

1127

172

171

1124

1128

192

1 106

189

190

1 109

"05

210

1088

207

208

I09I

1087

1122

'79

'77

1120

176

1 1 17

1111

182

184

"'3

185

1 1 16

1093

200

202

1095

203

1098

1009

287

286

1011

284

1014

ico8

290

291

1006

293

1003

990

308

309

988

3"

985

282

1016

280

279

1019

1015

295

IOOI

297

298

998

1002

3'3

983

3'5

3'6

980

984

276

275

1023

1024

1022

271

30'

302

994

993

995

306

3'9

320

976

975

977

324

102 1

1025

273

274

272

1026

996

992

3°4

303

305

99'

978

974

322

321

323

973

1020

278

1017

1018

281

'77

997

299

1000

999

296

300

979

3'7

982

981

3'4

3'8

283

1010

1012

285

1013

288

294

1007

1005

292

1004

289

312

989

987

3'0

986

3°7

S<"

935

934

363

932

366

360

938

939

358

94'

355

342

956

957

340

959

337

93°

368

928

927

371

367

943

353

945

946

350

354

961

335

963

964

332

336

924

923

375

376

374

919

949

95°

346

345

347

954

967

968

328

327

329

972

373

377

921

922

920

378

348

344

952

95'

953

343

330

326

970

969

97'

325

372

926

369

370

929

925

349

947

352

35'

944

948

33'

965

334

333

962

966

93'

362

3 6 4

933

365

936

942

359

357

940

356

937

960

34'

339

958

338

955

504

794

795

502

797

499

505

79'

790

5°7

788

510

523

773

772

525

77°

528

799

497

801

802

494

498

786

5'2

784

783

5'5

5"

768

53o

766

765

533

529

805

806

490

489

491

810

780

779

5'9

520

5'8

775

762

76i

537

538

536

757

492

488

808

807

809

487

5'7

52'

777

778

776

522

535

539

759

760

758

540

493

803

496

495

800

804

5'6

782

5'3

5'4

785

781

534

764

53'

532

767

763

798

503

5°'

70

500

79i

787

506

508

789

509

792

769

524

526

771

527

774

612

686

687

610

689

607

613

683

682

6'5

680

618

631

66 S

664

633

662

636

691

605

693

694

602

606

678

620

676

675

623

619

660

638

658

657

641

637

697

698

598

597

599

702

672

671

627

628

626

667

654

653

645

646

644

649

£00

595

700

699

701

595

625

629

669

670

668

630

643

647

651

652

650

648

601

695

604

603

692

696

624

674

621

622

677

673

642

656

639

640

659

655

690

611

609

688

608

685

679

614

616

681

617

684

661

632

634

663

635

666

Fig. 4, Second Part. (6':S=389i)

316

THE MONIST.

given in Magic Squares and Cubes and simply extend it to cover
the larger numbers involved we shall have a group of four cubes
in which all the "straight" dimensions have S=SI4, but no diag-
onals except the four cubic diagonals. Each horizontal row of
squares will produce a cube having exactly the same properties as
those in the four vertical rows. If the four squares in either diag-

I

409S

4094

4

5

4091

4090

8

4032

66

67

4029

4028

70

71

4025

4088

10

11

4085

4084

14

IS

4081

73

4023

4022

76

77

4019

4018

80

4080

18

19

4077

4076

22

23

4073

81

4015

4014

84

85

401 1

4010

88

25

4071

4070

28

29

4067

4066

32

4008

90

91

4005

4004

94

95

4001

406S

31

30

4068

4069

27

26

4072

96

4002

4003

93

92

4006

4007

89

24

4074

4075

31

20

4078

4079

17

4009

87

86

4012

4013

83

82

4016

16

4082

4083

13

12

4086

4087

9

4017

79

78

4020

4021

75

74

4024

4089

7

6

4092

4093

3

2

4096

72

4026

4027

69

68

4030

4031

65

4064

34

35

4061

4060

38

39

4057

97

3999

3098

100

101

399S

3994

104

41

4055

4054

44

45

4051

4050

48

3992

106

107

3989

3988

1 10

III

3985

49

4047

4046

52

53

4043

4042

56

3984

114

115

398i

3980

118

119

3977

4040

58

59

4037

4036

62

63

4033

121

3975

3974

124

125

3971

3970

128

64

4034

4035

61

60

4038

4039

57

3969

127

126

3972

3973

123

122

3976

4041

55

54

4044

4045

51

50

4048

120

3978

3979

117

116

3982

3983

113

4049

47

46

4052

4053

43

42

4056

112

3986

3987

109

108

3990

3991

ios

40

4058

4059

37

36

4062

4063

33

3993

103

ro2

3996

3997

99

98

4000

II IV

Fig. 5, 8*, First Part (One cube written).

onal of the figure be piled together neither vertical columns nor
cubic diagonals will have the correct summation, but all the diagonals
of vertical squares in either direction will. Regarding the whole
group of sixteen squares as a plane square we find it magic, having
the summation 2056 in every rank, file and corner diagonal, 1028

CRITICISMS AND DISCUSSIONS.

317

in each half-rank or half-file, and 514 in each quarter-rank or
quarter-file. Furthermore all complementary pairs are balanced about
the center.

The alternative arrangement shown in Fig. 3 makes each of the
small squares perfect in itself, with every rank, file and corner diag-
onal footing up 514 and complementary pairs balanced about the

3968

130

131

3965

39&t

134

135

396i

193

3903

3902

196

«97

3899

3898

200

137

3959

3958

140

141

3955

3954

144

3896

202

203

3893

3892

206

207

3889

145

3051

3950

148

149

3947

3946

152

3888

210

211

3885

3884

214

215

3881

3944

154

155

3941

3940

158

IS9

3937

217

3879

3878

220

221

3875

3874

224

160

3938

3939

157

156

3942

3943

153

3873

223

222

3876

3877

219

218

3880

3945

151

150

3948

3949

147

146

3952

216

38R?

3883

213

212

3886

3887

209

3953

143

142

3956

3957

139

138

3960

208

3890

3891

205

204

3894

3895

201

136

3962

3963

133

132

3966

3967

129

3897

199

198

3900

3901

195

194

3904

161

3935

3934

164

165

3931

393°

168

3872

226

227

3869

3868

230

231

3865

3928

170

171

3925

3924

»74

175

3921

233

3863

3862

236

237

3859

38S8

240

3920

178

179

35W7

39i6

182

183

3913

241

3855

3854

244

245

3851

3850

248

185

39"

39«o

188

«89

3907

3906

192

.1848

250

251

3845

3844

254

255

3841

3905

191

190

3908

3909

187

186

3912

256

3842

3843

253

252

3846

3847

249

184

3914

3915

181

180

3918

3919

177

3849

247

246

3852

3853

243

242

3856

176

3922

3923

173

172

3926

3927

169

3857

239

238

3860

3861

335

234

3864

3929

167

166

3932

3933

163

162

3936

232

3866

3867

229

228

3870

3871

225

VI VIII

Fig. 5, 8*, Second Part (One cube written).

center. As in the other arrangement the squares in each vertical
or horizontal row combine to make cubes whose "straight" dimen-
sions all have the right summation. In addition the new form has
the two plane diagonals of each original square (eight for each
cube), but sacrifices the four cubic diagonals in each cube. In lieu

318 THE MONIST.

of these we find a complete set of "bent diagonals" ("Franklin")
like those described for the magic cube of six in The Monist for
July, 1909.

If the four squares in either diagonal of the large figure be
piled up it will be found that neither cubic diagonal nor vertical
column is correct, but that all diagonals of vertical squares facing
toward front or back are. Taken as a plane figure the whole group
makes up a magic square of 16 with the summation 2056 in every
rank, file or corner diagonal, half that summation in half of each
of those dimensions, and one-fourth of it in each quarter dimension.

2. Let n=6, then 8=3891. — With the natural series 1. . .1296
squares were constructed which combined to produce the six magic
cubes of six indicated by the Roman numerals in Fig. 4. These
have all the characteristics of the 6-cube described in The Monist
of July last — 108 "straight" rows, 12 plane diagonals and 24 "bent"
diagonals in each cube, with the addition of 32 vertical-square
diagonals if the squares are piled in a certain order. A seventh
cube with the same features is made by combining the squares in
the lowest horizontal row — i. e., the bottom squares of the num-
bered cubes. The feature of the cubic bent diagonals is found on
combining any three of the small squares, no matter in what order
they are taken. In view of the recent discussion of this cube it seems
unnecessary to give any further account of it now.

The whole figure, made up as it is of thirty-six magic squares,
is itself a magic square of 36 with the proper summation (23346)
for every rank, file and corner diagonal, and the corresponding
fractional part of that for each half, third or sixth of those dimen-
sions. Any square group of four, nine, sixteen or twenty-five of the
small squares will be magic in all its dimensions.

3. Let n=8, then 8=16388. — The numbers 1...4096 may be
arranged in several different ways. If the diagrams in Mr. An-
drews's book be adopted we have a group of eight cubes in which
rank, file, column and cubic diagonal are correct (and in which
the halves of these dimensions have the half summation), but all
plane diagonals are irregular. If the plan be adopted of construct-
ing the small squares of complementary couplets, as in the 6-cube,
the plane diagonals are equalized at the cost of certain other features.
I have used therefore a plan which combines to some extent the ad-
vantages of both the others.

It will be noticed that each of the small squares in Fig. 5 is

CRITICISMS AND DISCUSSIONS. 319

perfect in that it has the summation 16388 for rank, file and corner
diagonal (also for broken diagonals if each of the separated parts
contain two, four or six — not an odd number of cells) , and in balan-
cing complementary couplets. When the eight squares are piled
one upon the other a cube results in which rank, file, column, the
plane diagonals of each horizontal square, the four ordinary cubic
diagonals and 32 cubic bent diagonals all have S=i6388. What is
still more remarkable, the half of each of the "straight" dimensions
and of each cubic diagonal has half that sum. Indeed this cube of
eight can be sliced into eight cubes of 4 in each of which every rank,
file, column and cubic diagonal has the footing 8194; and each of
these 4-cubes can be subdivided into eight tiny 2-cubes in each of
which the eight numbers foot up 16388.

So much for the features of the single cube here presented.
As a matter of fact only the one cube has actually been written out.
The plan of its construction, however, is so simple and the relations
of numbers so uniform in the powers of 8 that it was easy to in-
vestigate the properties of the whole 8 4 scheme without having the
squares actually before me. I give here the initial number of each
of the eight squares in each of the eight cubes, leaving it for some
one possessed of more leisure to write them all out and verify my
statements as to the intercubical features. It should be remembered
that in each square the number diagonally opposite the one here
given is its complement, i. e., the number which added to it will
give the sum 4097.

I

II

III

IV

V

VI

VII

VIII

I

3840

3584

769

3072

1281

1537

2304

4064

289

545

3296

1057

2784

2528

1825

4032

321

577

3264

1089

2752

2496

1857

97

3744

3488

865

2976

1377

1633

2208

3968

385

641

3200

1 153

2688

2432

1921

161

3680

3424

929

2912

1441

1697

2144

193

3648

3392

961

2880

1473

1729

2112

3872

481

737

3104

1249

2592

2336

2017

16388 16388 16388 16388 16388 16388 16388 16388

Each of the sixty-four numbers given above will be at the
upper left-hand corner of a square and its complement at the lower
right-hand corner. The footings given are for these initial numbers,

320 THE MONIST.

but the arrangement of numbers in the squares is such that the
footing will be the same for every one of the sixty-four columns
in each cube. If the numbers in each horizontal line of the table
above be added they will be found to have the same sum: conse-
quently the squares headed by them must make a cube as nearly
perfect as the example given in Fig. 5, which is cube I of the table
above. But the sum of half the numbers in each line is half of
16388, and hence each of the eight cubes formed by taking the
squares in the horizontal rows is capable of subdivision into 4-cubes
and 2-cubes, like our original cube. We thus have sixteen cubes,
each with the characteristics described for the one presented in

Fig- 5-

If we pile the squares lying in the diagonal of our great square
(starting with 1, 289, etc., or 2304, 2528, etc.) we find that its col-
umns and cubic diagonals are not correct; but all the diagonals of
its vertical squares are so, and even here the remarkable feature of
the half-dimension persists.

Of course there is nothing to prevent one's going still further
and examining constructions involving the fifth or even higher pow-
ers, but the utility of such research may well be doubted. The purpose